论文标题
单调变异不平等的基于近似的正则额外梯度法
An Approximation-Based Regularized Extra-Gradient Method for Monotone Variational Inequalities
论文作者
论文摘要
在本文中，我们提出了一种解决单调变化不平等（VI）的一般额外梯度方案，此处称为基于近似的正则化额外梯度法（AS）。第一步是解决VI子问题，其中一个近似操作员满足了$ p^{th} $ - 订购Lipschitz相对于原始映射的绑定，并与$（p+1）^{th} $的梯度进一步相结合。通过包括一个额外的额外梯度步骤，可以保证最佳的全局收敛性，而如果VI强烈单调，则显示出$ p^{th} $ - 订购超级线性本地收敛。提出的是包容性和一般性的，从某种意义上说，可以在本框架内提出各种解决方案方法，因为近似值的不同表现形式，其迭代复杂性将以统一的方式遵循。该框架与一阶方法有关，同时为专门针对确保最佳迭代复杂性界限的结构化问题开发了高阶方法的可能性。
In this paper, we propose a general extra-gradient scheme for solving monotone variational inequalities (VI), referred to here as Approximation-based Regularized Extra-gradient method (ARE). The first step of ARE solves a VI subproblem with an approximation operator satisfying a $p^{th}$-order Lipschitz bound with respect to the original mapping, further coupled with the gradient of a $(p+1)^{th}$-order regularization. The optimal global convergence is guaranteed by including an additional extra-gradient step, while a $p^{th}$-order superlinear local convergence is shown to hold if the VI is strongly monotone. The proposed ARE is inclusive and general, in the sense that a variety of solution methods can be formulated within this framework as different manifestations of approximations, and their iteration complexities would follow through in a unified fashion. The ARE framework relates to the first-order methods, while opening up possibilities to developing higher-order methods specifically for structured problems that guarantee the optimal iteration complexity bounds.
